3.2517 \(\int \frac{(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{376 (8 x+7)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(-2*(3 + 2*x)^2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (376*(7 + 8*x))/(3*Sq
rt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.0861762, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{376 (8 x+7)}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 (2 x+3)^2 (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^2)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(3 + 2*x)^2*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (376*(7 + 8*x))/(3*Sq
rt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 13.4927, size = 48, normalized size = 0.89 \[ - \frac{2 \left (2 x + 3\right )^{2} \left (35 x + 29\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{188 \left (16 x + 14\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**2/(3*x**2+5*x+2)**(5/2),x)

[Out]

-2*(2*x + 3)**2*(35*x + 29)/(3*(3*x**2 + 5*x + 2)**(3/2)) + 188*(16*x + 14)/(3*s
qrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.0478535, size = 33, normalized size = 0.61 \[ \frac{2 \left (4372 x^3+10932 x^2+8925 x+2371\right )}{3 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^2)/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(2*(2371 + 8925*x + 10932*x^2 + 4372*x^3))/(3*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.007, size = 38, normalized size = 0.7 \[{\frac{ \left ( 8744\,{x}^{3}+21864\,{x}^{2}+17850\,x+4742 \right ) \left ( 1+x \right ) \left ( 2+3\,x \right ) }{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(5/2),x)

[Out]

2/3*(4372*x^3+10932*x^2+8925*x+2371)*(1+x)*(2+3*x)/(3*x^2+5*x+2)^(5/2)

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Maxima [A]  time = 0.714355, size = 103, normalized size = 1.91 \[ \frac{8744 \, x}{9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{4 \, x^{2}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} + \frac{21860}{27 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{1114 \, x}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{1042}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^2*(x - 5)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="maxima")

[Out]

8744/9*x/sqrt(3*x^2 + 5*x + 2) + 4/3*x^2/(3*x^2 + 5*x + 2)^(3/2) + 21860/27/sqrt
(3*x^2 + 5*x + 2) - 1114/27*x/(3*x^2 + 5*x + 2)^(3/2) - 1042/27/(3*x^2 + 5*x + 2
)^(3/2)

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Fricas [A]  time = 0.272867, size = 69, normalized size = 1.28 \[ \frac{2 \,{\left (4372 \, x^{3} + 10932 \, x^{2} + 8925 \, x + 2371\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{3 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^2*(x - 5)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="fricas")

[Out]

2/3*(4372*x^3 + 10932*x^2 + 8925*x + 2371)*sqrt(3*x^2 + 5*x + 2)/(9*x^4 + 30*x^3
+ 37*x^2 + 20*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{51 x}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{8 x^{2}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{4 x^{3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{45}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**2/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-51*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2)
+ 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 +
5*x + 2)), x) - Integral(-8*x**2/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(
3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2)
+ 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(4*x**3/(9*x**4*sqrt(3*x**2 + 5*x + 2
) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(
3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-45/(9*x**4*sqrt(3*
x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2
) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

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GIAC/XCAS [A]  time = 0.279135, size = 38, normalized size = 0.7 \[ \frac{2 \,{\left ({\left (4 \,{\left (1093 \, x + 2733\right )} x + 8925\right )} x + 2371\right )}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^2*(x - 5)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="giac")

[Out]

2/3*((4*(1093*x + 2733)*x + 8925)*x + 2371)/(3*x^2 + 5*x + 2)^(3/2)